MathDB

2

Part of 2007 District Olympiad

Problems(6)

Game with balls

Source: Romanian DMO, 7th grade, problem 2

3/23/2007
In an urn we have red and blue balls. A person has invented the next game: he extracts balls until he realises for the first time that the number of blue balls is equal to the number of red balls. After a such game he finds out that he has extracted 10 balls, and that there does not exist 3 consecutive balls of the same color. Prove that the fifth and the sixth balls have different collors.
combinatorics proposedcombinatorics
midpoint and perpendicular wanted in 3D, planes angle given, rectangle

Source: 2007 Romania District VIII P2

5/18/2020
Consider a rectangle ABCDABCD with AB=2AB = 2 and BC=3BC = \sqrt3. The point MM lies on the side ADAD so that MD=2AMMD = 2 AM and the point NN is the midpoint of the segment ABAB. On the plane of the rectangle rises the perpendicular MP and we choose the point QQ on the segment MPMP such that the measure of the angle between the planes (MPC)(MPC) and (NPC)(NPC) shall be 45o45^o, and the measure of the angle between the planes (MPC)(MPC) and (QNC)(QNC) shall be 60o60^o.
a) Show that the lines DNDN and CMCM are perpendicular.
b) Show that the point QQ is the midpoint of the segment MPMP.
geometry3D geometryrectangleanglesrectnagleperpendicular bisectormidpoint
Straightforward vectors

Source: RMO District Round, 9th grade, 2007

1/21/2008
Consider △ABC \triangle ABC and points M∈(AB) M \in (AB), N∈(BC) N \in (BC), P∈(CA) P \in (CA), R∈(MN) R \in (MN), S∈(NP) S \in (NP), T∈(PM) T \in (PM) such that \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k and \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k for some k∈(0,1) k \in (0, 1). Prove that △STR∼△ABC \triangle STR \sim \triangle ABC and, furthermore, determine k k for which the minimum of [STR] [STR] is attained.
vectorgeometry proposedgeometry
Romania District Olympiad 2007 - Grade XI

Source:

4/10/2011
Let A∈Mn(R∗)A\in \mathcal{M}_n(\mathbb{R}^*). If A⋅ tA=InA\cdot\ ^t A=I_n, prove that:
a)∣Tr(A)∣≤n|\text{Tr}(A)|\le n;
b)If nn is odd, then det⁡(A2−In)=0\det(A^2-I_n)=0.
linear algebralinear algebra unsolved
Record number of integral signs in a inequality

Source: RMO 2007 - District Round - II

3/3/2007
Let f:[0,1]→Rf : \left[ 0, 1 \right] \to \mathbb R be a continuous function and g:[0,1]→(0,∞)g : \left[ 0, 1 \right] \to \left( 0, \infty \right). Prove that if ff is increasing, then ∫0tf(x)g(x) dx⋅∫01g(x) dx≤∫0tg(x) dx⋅∫01f(x)g(x) dx.\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .
calculusintegrationinequalitiesfunctionderivativeprobabilityexpected value
2xn tablet colored with three colors

Source: Romania District Olympiad 2007, Grade X, Problem 2

10/7/2018
All 2n≥2 2n\ge 2 squares of a 2×n 2\times n rectangle are colored with three colors. We say that a color has a cut if there is some column (from all n n ) that has both squares colored with it. Determine:
a) the number of colorings that have no cuts. b) the number of colorings that have a single cut.
geometryrectanglecombinatoricsnumberingcounting